Show that 27 x 23^n + 17 x 10^2n is divisible by 11 for all positive integers n. I know that modulos should aid in answering this problem,

Answer 1

27= 2 mode 5
23= 2 mode 1
17= 1 mode 6
10^2n= 100^n
100= 1 mode 9
So the problem reduces to
5*1^n+6*1^n
=5+6
=11

Answer 2

oops,typo error.

100=9 mode 1 the rest is correct.